Sum of factorials for large numbers

问题:

I want to calculate the sum of digits of N!.

I want to do this for really large values of N, say N(1500). I am not using .NET 4.0. I cannot use the BigInteger class to solve this.

Can this be solved by some other algorithm or procedure? Please help.

I want to do some thing like this Calculate the factorial of an arbitrarily large number, showing all the digits but in C#. However I am unable to solve.

回答1:

There is no special magic that allows you to calculate the sum of the digits, as far as I am concerned.

It shouldn't be that hard to create your own BigInteger class anyway - you only need to implement the long multiplication algorithm from 3rd grade.

回答2:

If your goal is to calculate the sum of the digits of N!, and if N is reasonably bounded, you can do the following without a BigInteger type:

• Find a list of factorial values online (table lookup will be much more efficient than calculating from scratch, and does not require BigInteger)
• Store as a string data type
• Parse each character in the string as an integer
• Add the resulting integers

回答3:

There are two performance shortcuts that you can use for whatever implementation you choose.

1. Chop off any zeros from the numbers.
2. If the number is evenly divisible by 5^n, divide it by 10^n.

in this way,

16*15*14*13*12*11*10*9*8*7*6*5*4*3*2 = 20,922,789,888,000
//-->
16*1.5*14*13*12*11*1*9*8*7*6*0.5*4*3*2 = 20,922,789,888 //Sum of 63

---

Also, it feels like there should be some algorithm without reverting to calculating it all out. Going to 18!, the sums of the digits are:

2,6,6,3,9,9,9,27,27,36,27,27,45,45,63,63,63
//the sums of the resulting digits are:
2,6,6,3,9,9,9,9,9,9,9,9,9,9,9,9,9

and notably, the sum of the digits of 1500! is 16749 (the sum of whose digits are 27)

回答4:

Here's some working code. Some components can be improved upon to increase efficiency. The idea is to use whatever multiplication algorithm I was told in school, and to store long integers as strings.

As an afterthought, I think it would be smarter to represent large numbers with List<int>() instead of string. But I'll leave that as an exercise to the reader.

Code Sample

static string Mult(string a, string b)
    {
        int shift = 0;
        List<int> result = new List<int>();
        foreach (int aDigit in a.Reverse().Select(c => int.Parse(c.ToString())))
        {
            List<int> subresult = new List<int>();
            int store = 0;
            foreach (int bDigit in b.Reverse().Select(c => int.Parse(c.ToString())))
            {
                int next = aDigit*bDigit + store;
                subresult.Add(next%10);
                store = next/10;
            }

            if (store != 0) subresult.Add(store);

            subresult.Reverse();
            for (int i = 0; i < shift; ++i) subresult.Add(0);
            subresult.Reverse();

            int newResult = new List<int>();
            store = 0;
            for (int i = 0; i < subresult.Count; ++i)
                {
                    if (result.Count >= i + 1)
                    {
                        int next = subresult[i] + result[i] + store;
                        if (next >= 10)
                            newResult.Add(next % 10);
                        else newResult.Add(next);
                        store = next / 10;
                    }
                    else
                    {
                        int next = subresult[i] + store;
                        newResult.Add(next % 10);
                        store = next / 10;
                    }
                }

            if (store != 0) newResult.Add(store);

            result = newResult;
            ++shift;
        }

        result.Reverse();
        return string.Join("", result);
    }

    static int FactorialSum(int n)
    {
        string result = "1";
        for (int i = 2; i <= n; i++)
            result = Mult(i.ToString(), result);
        return result.Sum(r => int.Parse(r.ToString()));
    }

Code Testing

Assuming the code snippet above is in the same class as your Main method, call it thusly.

Input

    static void Main(string[] args)
    {
        Console.WriteLine(FactorialSum(1500));
    }

Output

16749

回答5:

Here's a port of the C++ code you reference in one of your comments. One thing to realize when porting from C++ to C# is that integers that are zero evaluate to false and integers that are non-zero evaluate to true when used in a Boolean comparison.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace ArbitraryFactorial
{
    class Program
    {
        const int max = 5000;

        static void display(int[] arr)
        {
            int ctr = 0;
            for (int i = 0; i < max; i++)
            {
                if (ctr == 0 && arr[i] != 0) ctr = 1;
                if (ctr != 0)
                    Console.Write(arr[i]);

            }
        }

        static void factorial(int[] arr, int n)
        {
            if (n == 0) return;
            int carry = 0;
            for (int i = max - 1; i >= 0; --i)
            {
                arr[i] = (arr[i] * n) + carry;
                carry = arr[i] / 10;
                arr[i] %= 10;
            }
            factorial(arr, n - 1);
        }

        static void Main(string[] args)
        {
            int[] arr = new int[max];
            arr[max - 1] = 1;
            int num;
            Console.Write("Enter the number: ");
            num = int.Parse(Console.ReadLine());
            Console.Write("Factorial of " + num + " is: ");
            factorial(arr, num);
            display(arr);
        }
    }
}

回答6:

you can find the source code at : http://codingloverlavi.blogspot.in/2013/03/here-is-one-more-interesting-program.html

#include<stdio.h>
#include<conio.h>
#include<iostream.h>
#include<time.h>
#define max 5000
void multiply(long int *,long int);
void factorial(long int *,long int);


int main()
{
clrscr();
cout<<"PROGRAM TO CALCULATE FACTORIAL OF A NUMBER";
cout<<"\nENTER THE NUMBER\n";
long int num;
cin>>num;

long int a[max];
for(long int i=0;i<max;i++)
    a[i]=0;

factorial(a,num);

clrscr();

//PRINTING THE FINAL ARRAY...:):):)
cout<<"THE FACTORIAL OF "<<num<<" is "<<endl<<endl;
long int flag=0;

int ans=0;
for(i=0;i<max;i++)
{
    if(flag||a[i]!=0)
    {
        flag=1;
        cout<<a[i];
        ans=ans+a[i];
    }
}

cout<<endl<<endl<<"the sum of all digits is: "<<ans;


getch();
return 1;
}

void factorial(long int *a,long int n)
{
long int lavish;
long int num=n;
lavish=n;
for(long int i=max-1;i>=0&&n;i--)
{
    a[i]=n%10;
    n=n/10;
}

for(i=2;i<(lavish);i++)
{
    multiply(a,num-1);
    num=num-1;

}
}


void multiply(long int *a,long int n)
{

for(long int i=0;i<max;i++)
    a[i]=a[i]*n;

for(i=max-1;i>0;i--)
{
    a[i-1]=a[i-1]+(a[i]/10);
    a[i]=a[i]%10;
}
}

回答7:

You can't use these numbers at all without a BigInteger type.
No algorithm or procedure can squeeze numbers larger than 2⁶⁴ into a long.

You need to find a BigInteger implementation for .Net 3.5.